Definitions
Lorentz force on a charged particle (of charge q ) in motion (velocity v ), used as the definition of the E field and B field . Here subscripts e and m are used to differ between electric and magnetic charges . The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by qm (Wb) = μ 0 qm (Am).
Initial quantities Electric quantities Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂ , d is the dipole moment between two point charges, the volume density of these is the polarization density P . Position vector r is a point to calculate the electric field; r′ is a point in the charged object. Contrary to the strong analogy between (classical) gravitation and electrostatics , there are no "centre of charge" or "centre of electrostatic attraction" analogues.
Electric transport
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Linear, surface, volumetric charge density λe for Linear, σe for surface, ρe for volume. q e = ∫ λ e d ℓ {\displaystyle q_{e}=\int \lambda _{e}\mathrm {d} \ell } q e = ∬ σ e d S {\displaystyle q_{e}=\iint \sigma _{e}\mathrm {d} S}
q e = ∭ ρ e d V {\displaystyle q_{e}=\iiint \rho _{e}\mathrm {d} V}
C m−n , n = 1, 2, 3 [I][T][L]−n Capacitance C C = d q / d V {\displaystyle C=\mathrm {d} q/\mathrm {d} V\,\!} V = voltage, not volume.
F = C V−1 [I]2 [T]4 [L]−2 [M]−1 Electric current I I = d q / d t {\displaystyle I=\mathrm {d} q/\mathrm {d} t\,\!} A [I] Electric current density J I = J ⋅ d S {\displaystyle I=\mathbf {J} \cdot \mathrm {d} \mathbf {S} } A m−2 [I][L]−2 Displacement current density J d J d = ϵ 0 ( ∂ E / ∂ t ) = ∂ D / ∂ t {\displaystyle \mathbf {J} _{\mathrm {d} }=\epsilon _{0}\left(\partial \mathbf {E} /\partial t\right)=\partial \mathbf {D} /\partial t\,\!} A m−2 [I][L]−2 Convection current density J c J c = ρ v {\displaystyle \mathbf {J} _{\mathrm {c} }=\rho \mathbf {v} \,\!} A m−2 [I][L]−2
Electric fields
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Electric field , field strength, flux density, potential gradient E E = F / q {\displaystyle \mathbf {E} =\mathbf {F} /q\,\!} N C−1 = V m−1 [M][L][T]−3 [I]−1 Electric flux ΦE Φ E = ∫ S E ⋅ d A {\displaystyle \Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!} N m2 C−1 [M][L]3 [T]−3 [I]−1 Absolute permittivity ; ε ϵ = ϵ r ϵ 0 {\displaystyle \epsilon =\epsilon _{r}\epsilon _{0}\,\!} F m−1 [I]2 [T]4 [M]−1 [L]−3 Electric dipole moment p p = q a {\displaystyle \mathbf {p} =q\mathbf {a} \,\!} a = charge separation directed from -ve to +ve charge
C m [I][T][L] Electric Polarization, polarization density P P = d ⟨ p ⟩ / d V {\displaystyle \mathbf {P} =\mathrm {d} \langle \mathbf {p} \rangle /\mathrm {d} V\,\!} C m−2 [I][T][L]−2 Electric displacement field , flux density D D = ϵ E = ϵ 0 E + P {\displaystyle \mathbf {D} =\epsilon \mathbf {E} =\epsilon _{0}\mathbf {E} +\mathbf {P} \,} C m−2 [I][T][L]−2 Electric displacement flux ΦD Φ D = ∫ S D ⋅ d A {\displaystyle \Phi _{D}=\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} \,\!} C [I][T] Absolute electric potential , EM scalar potential relative to point r 0 {\displaystyle r_{0}\,\!} Theoretical: r 0 = ∞ {\displaystyle r_{0}=\infty \,\!} Practical: r 0 = R e a r t h {\displaystyle r_{0}=R_{\mathrm {earth} }\,\!} (Earth's radius)
φ ,V V = − W ∞ r q = − 1 q ∫ ∞ r F ⋅ d r = − ∫ r 1 r 2 E ⋅ d r {\displaystyle V=-{\frac {W_{\infty r}}{q}}=-{\frac {1}{q}}\int _{\infty }^{r}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 Voltage , Electric potential difference Δφ ,ΔV Δ V = − Δ W q = − 1 q ∫ r 1 r 2 F ⋅ d r = − ∫ r 1 r 2 E ⋅ d r {\displaystyle \Delta V=-{\frac {\Delta W}{q}}=-{\frac {1}{q}}\int _{r_{1}}^{r_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1
Magnetic quantities Magnetic transport
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Linear, surface, volumetric pole density λm for Linear, σm for surface, ρm for volume. q m = ∫ λ m d ℓ {\displaystyle q_{m}=\int \lambda _{m}\mathrm {d} \ell } q m = ∬ σ m d S {\displaystyle q_{m}=\iint \sigma _{m}\mathrm {d} S}
q m = ∭ ρ m d V {\displaystyle q_{m}=\iiint \rho _{m}\mathrm {d} V}
Wb m−n A m(−n + 1) , n = 1, 2, 3
[L]2 [M][T]−2 [I]−1 (Wb) [I][L] (Am)
Monopole current Im I m = d q m / d t {\displaystyle I_{m}=\mathrm {d} q_{m}/\mathrm {d} t\,\!} Wb s−1 A m s−1
[L]2 [M][T]−3 [I]−1 (Wb) [I][L][T]−1 (Am)
Monopole current density J m I = ∬ J m ⋅ d A {\displaystyle I=\iint \mathbf {J} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {A} } Wb s−1 m−2 A m−1 s−1
[M][T]−3 [I]−1 (Wb) [I][L]−1 [T]−1 (Am)
Magnetic fields
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Magnetic field , field strength, flux density, induction field B F = q e ( v × B ) {\displaystyle \mathbf {F} =q_{e}\left(\mathbf {v} \times \mathbf {B} \right)\,\!} T = N A−1 m−1 = Wb m−2 [M][T]−2 [I]−1 Magnetic potential , EM vector potential A B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } T m = N A−1 = Wb m3 [M][L][T]−2 [I]−1 Magnetic flux ΦB Φ B = ∫ S B ⋅ d A {\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} \,\!} Wb = T m2 [L]2 [M][T]−2 [I]−1 Magnetic permeability μ {\displaystyle \mu \,\!} μ = μ r μ 0 {\displaystyle \mu \ =\mu _{r}\,\mu _{0}\,\!} V·s·A−1 ·m−1 = N·A−2 = T·m·A−1 = Wb·A−1 ·m−1 [M][L][T]−2 [I]−2 Magnetic moment , magnetic dipole moment m , μB , Π Two definitions are possible:
using pole strengths, m = q m a {\displaystyle \mathbf {m} =q_{m}\mathbf {a} \,\!}
using currents: m = N I A n ^ {\displaystyle \mathbf {m} =NIA\mathbf {\hat {n}} \,\!}
a = pole separation
N is the number of turns of conductor
A m2 [I][L]2 Magnetization M M = d ⟨ m ⟩ / d V {\displaystyle \mathbf {M} =\mathrm {d} \langle \mathbf {m} \rangle /\mathrm {d} V\,\!} A m−1 [I] [L]−1 Magnetic field intensity, (AKA field strength) H Two definitions are possible: most common: B = μ H = μ 0 ( H + M ) {\displaystyle \mathbf {B} =\mu \mathbf {H} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,}
using pole strengths, H = F / q m {\displaystyle \mathbf {H} =\mathbf {F} /q_{m}\,}
A m−1 [I] [L]−1 Intensity of magnetization , magnetic polarization I , J I = μ 0 M {\displaystyle \mathbf {I} =\mu _{0}\mathbf {M} \,\!} T = N A−1 m−1 = Wb m−2 [M][T]−2 [I]−1 Self Inductance L Two equivalent definitions are possible: L = N ( d Φ / d I ) {\displaystyle L=N\left(\mathrm {d} \Phi /\mathrm {d} I\right)\,\!}
L ( d I / d t ) = − N V {\displaystyle L\left(\mathrm {d} I/\mathrm {d} t\right)=-NV\,\!}
H = Wb A−1 [L]2 [M] [T]−2 [I]−2 Mutual inductance M Again two equivalent definitions are possible: M 1 = N ( d Φ 2 / d I 1 ) {\displaystyle M_{1}=N\left(\mathrm {d} \Phi _{2}/\mathrm {d} I_{1}\right)\,\!}
M ( d I 2 / d t ) = − N V 1 {\displaystyle M\left(\mathrm {d} I_{2}/\mathrm {d} t\right)=-NV_{1}\,\!}
1,2 subscripts refer to two conductors/inductors mutually inducing voltage/ linking magnetic flux through each other. They can be interchanged for the required conductor/inductor;
M 2 = N ( d Φ 1 / d I 2 ) {\displaystyle M_{2}=N\left(\mathrm {d} \Phi _{1}/\mathrm {d} I_{2}\right)\,\!} M ( d I 1 / d t ) = − N V 2 {\displaystyle M\left(\mathrm {d} I_{1}/\mathrm {d} t\right)=-NV_{2}\,\!}
H = Wb A−1 [L]2 [M] [T]−2 [I]−2 Gyromagnetic ratio (for charged particles in a magnetic field) γ ω = γ B {\displaystyle \omega =\gamma B\,\!} Hz T−1 [M]−1 [T][I]
Electric circuits DC circuits, general definitions
AC circuits
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Resistive load voltage VR V R = I R R {\displaystyle V_{R}=I_{R}R\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 Capacitive load voltage VC V C = I C X C {\displaystyle V_{C}=I_{C}X_{C}\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 Inductive load voltage VL V L = I L X L {\displaystyle V_{L}=I_{L}X_{L}\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 Capacitive reactance XC X C = 1 ω d C {\displaystyle X_{C}={\frac {1}{\omega _{\mathrm {d} }C}}\,\!} Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2 Inductive reactance XL X L = ω d L {\displaystyle X_{L}=\omega _{d}L\,\!} Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2 AC electrical impedance Z V = I Z {\displaystyle V=IZ\,\!} Z = R 2 + ( X L − X C ) 2 {\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}\,\!}
Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2 Phase constant δ, φ tan ϕ = X L − X C R {\displaystyle \tan \phi ={\frac {X_{L}-X_{C}}{R}}\,\!} dimensionless dimensionless AC peak current I 0 I 0 = I r m s 2 {\displaystyle I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!} A [I] AC root mean square current I rms I r m s = 1 T ∫ 0 T [ I ( t ) ] 2 d t {\displaystyle I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!} A [I] AC peak voltage V 0 V 0 = V r m s 2 {\displaystyle V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 AC root mean square voltage V rms V r m s = 1 T ∫ 0 T [ V ( t ) ] 2 d t {\displaystyle V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t}}\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 AC emf, root mean square E r m s , ⟨ E ⟩ {\displaystyle {\mathcal {E}}_{\mathrm {rms} },{\sqrt {\langle {\mathcal {E}}\rangle }}\,\!} E r m s = E m / 2 {\displaystyle {\mathcal {E}}_{\mathrm {rms} }={\mathcal {E}}_{\mathrm {m} }/{\sqrt {2}}\,\!} V = J C−1 [M] [L]2 [T]−3 [I]−1 AC average power ⟨ P ⟩ {\displaystyle \langle P\rangle \,\!} ⟨ P ⟩ = E I r m s cos ϕ {\displaystyle \langle P\rangle ={\mathcal {E}}I_{\mathrm {rms} }\cos \phi \,\!} W = J s−1 [M] [L]2 [T]−3 Capacitive time constant τC τ C = R C {\displaystyle \tau _{C}=RC\,\!} s [T] Inductive time constant τL τ L = L / R {\displaystyle \tau _{L}=L/R\,\!} s [T]
Magnetic circuits Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Magnetomotive force , mmf F , F , M {\displaystyle {\mathcal {F}},{\mathcal {M}}} M = N I {\displaystyle {\mathcal {M}}=NI} N = number of turns of conductor
A [I]
Electromagnetism
Electric fields General Classical Equations
Magnetic fields and moments General classical equations
Electric circuits and electronics
Below N = number of conductors or circuit components. Subscript net refers to the equivalent and resultant property value.
Physical situation Nomenclature Series Parallel Resistors and conductors Ri = resistance of resistor or conductor i Gi = conductance of resistor or conductor i R n e t = ∑ i = 1 N R i {\displaystyle R_{\mathrm {net} }=\sum _{i=1}^{N}R_{i}\,\!} 1 G n e t = ∑ i = 1 N 1 G i {\displaystyle {1 \over G_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over G_{i}}\,\!}
1 R n e t = ∑ i = 1 N 1 R i {\displaystyle {1 \over R_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over R_{i}}\,\!} G n e t = ∑ i = 1 N G i {\displaystyle G_{\mathrm {net} }=\sum _{i=1}^{N}G_{i}\,\!}
Charge, capacitors, currents Ci = capacitance of capacitor i qi = charge of charge carrier i q n e t = ∑ i = 1 N q i {\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!} 1 C n e t = ∑ i = 1 N 1 C i {\displaystyle {1 \over C_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over C_{i}}\,\!} I n e t = I i {\displaystyle I_{\mathrm {net} }=I_{i}\,\!}
q n e t = ∑ i = 1 N q i {\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!} C n e t = ∑ i = 1 N C i {\displaystyle C_{\mathrm {net} }=\sum _{i=1}^{N}C_{i}\,\!} I n e t = ∑ i = 1 N I i {\displaystyle I_{\mathrm {net} }=\sum _{i=1}^{N}I_{i}\,\!}
Inductors Li = self-inductance of inductor i Lij = self-inductance element ij of L matrix Mij = mutual inductance between inductors i and j L n e t = ∑ i = 1 N L i {\displaystyle L_{\mathrm {net} }=\sum _{i=1}^{N}L_{i}\,\!} 1 L n e t = ∑ i = 1 N 1 L i {\displaystyle {1 \over L_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over L_{i}}\,\!} V i = ∑ j = 1 N L i j d I j d t {\displaystyle V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j}}{\mathrm {d} t}}\,\!}
Circuit DC Circuit equations AC Circuit equations Series circuit equations RC circuits Circuit equation R d q d t + q C = E {\displaystyle R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!}
Capacitor charge q = C E ( 1 − e − t / R C ) {\displaystyle q=C{\mathcal {E}}\left(1-e^{-t/RC}\right)\,\!}
Capacitor discharge q = C E e − t / R C {\displaystyle q=C{\mathcal {E}}e^{-t/RC}\,\!}
RL circuits Circuit equation L d I d t + R I = E {\displaystyle L{\frac {\mathrm {d} I}{\mathrm {d} t}}+RI={\mathcal {E}}\,\!}
Inductor current rise I = E R ( 1 − e − R t / L ) {\displaystyle I={\frac {\mathcal {E}}{R}}\left(1-e^{-Rt/L}\right)\,\!}
Inductor current fall I = E R e − t / τ L = I 0 e − R t / L {\displaystyle I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-Rt/L}\,\!}
LC circuits Circuit equation L d 2 q d t 2 + q / C = E {\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\,\!}
Circuit equation L d 2 q d t 2 + q / C = E sin ( ω 0 t + ϕ ) {\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!}
Circuit resonant frequency ω r e s = 1 / L C {\displaystyle \omega _{\mathrm {res} }=1/{\sqrt {LC}}\,\!}
Circuit charge q = q 0 cos ( ω t + ϕ ) {\displaystyle q=q_{0}\cos(\omega t+\phi )\,\!}
Circuit current I = − ω q 0 sin ( ω t + ϕ ) {\displaystyle I=-\omega q_{0}\sin(\omega t+\phi )\,\!}
Circuit electrical potential energy U E = q 2 / 2 C = Q 2 cos 2 ( ω t + ϕ ) / 2 C {\displaystyle U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!}
Circuit magnetic potential energy U B = Q 2 sin 2 ( ω t + ϕ ) / 2 C {\displaystyle U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!}
RLC Circuits Circuit equation L d 2 q d t 2 + R d q d t + q C = E {\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!}
Circuit equation L d 2 q d t 2 + R d q d t + q C = E sin ( ω 0 t + ϕ ) {\displaystyle L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!}
Circuit charge
q = q 0 e T − R t / 2 L cos ( ω ′ t + ϕ ) {\displaystyle q=q_{0}eT^{-Rt/2L}\cos(\omega 't+\phi )\,\!}
See also
Sources
Further reading
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